Alternatively, the thermal absorbing material may be internal to the hotspots. In this case, assuming an electron temperature of 8000 K and a hotspot diameter of 2 kpc [1]}, a particle density of about \(n_e = 2\)  cm\(^{-3}\)  would be required. Such high-density thermal material in the hotspots would imply high internal rotation measures. Indeed, [2]} obtained an upper limit to the hotspot thermal plasma density of \(n_e < 4 \times 10^{-4}\)  cm\(^{-3}\)  based on the low Faraday depolarisation observed at GHz frequencies. They suggest that complex models of magnetic field structure may hide the thermal material to some extent, and increase the limit on the thermal plasma density by up to two orders of magnitude, but even then, the upper limit would still be two orders of magnitude below the density required by the free-free absorption model (see also [3]}). For this reason, the free-free absorption model alone also does not currently provide a realistic explanation to account for the turnover in the radio spectra of the hotspots that is observed with LOFAR.
